Chemical Kinetics

Chemical Kinetics

It is the branch of physical chemistry that deals with the velocity or rate as well as the actual mechanism of the reaction.

Concept of rate of reaction

The average rate or rate of reaction is defined as the change in concentration of either reactant or product per unit time.

Let us suppose the following hypothetical reaction:
A(reactant) → B(product)

The average rate or rate of reaction is expressed either by the following ways in terms of reactant or product.

i. Rate of disappearance or decrease in the concentration of reactant:

Rate of reaction= (Decrease in conc. of reactant/Time taken)

ii. Rate of appearance or increase in the concentration of the product:

Rate of reaction= (Increase in conc. of product/Time taken)

Symbolically change in concentration of either reactant or product is represented by enclosing a square bracket of reacting species where square bracket represents the concentration in terms of mole per litre.

Rate\ of\ rxn=-\frac{\Delta[A]}{\Delta t}=+\frac{\Delta[B]}{\Delta t}

where ΔA is the concentration of reactant
ΔB is the concentration of product
Δt is the time taken.

In a reaction,
aA + bB → cC + dD

The average rate for this reaction is represented as:

\frac{\Delta x}{\Delta t}=-\frac{1}{a}\ \frac{\Delta [A]}{\Delta t}=-\frac{1}{b}\ \frac{\Delta [B]}{\Delta t}=+\frac{1}{c}\ \frac{\Delta [C]}{\Delta t}=+\frac{1}{d}\ \frac{\Delta [D]}{\Delta t}

Example:

N₂ + 3H₂ → 2NH₃

i. Rate of formation of ammonia = Δ[NH₃] / Δt
ii. Rate of disappearance of nitrogen = Δ[N₂] / Δt
iii. Rate of disappearance of hydrogen = Δ[H₂] / Δt

The rate of reaction is:

-\frac{\Delta [N_{2}]}{\Delta t}=-\frac{1}{3}\ \frac{\Delta [H_{2}]}{\Delta t}=+\frac{1}{2}\ \frac{\Delta [NH_{3}]}{\Delta t}

Instantaneous rate of reaction

It is defined as the change in concentration of reactant or product at a particular instant of time. It is denoted by dx/dt.
For a general reaction:
aA + bB → cC + dD

\frac{dx}{dt}=-\frac{1}{a}\ \frac{d[A]}{dt}=-\frac{1}{b}\ \frac{d[B]}{dt}=+\frac{1}{c}\ \frac{d[C]}{dt}=+\frac{1}{d}\ \frac{d[D]}{dt}

The unit of rate of reaction is mole L^-1 sec^-1

Factors affecting rate of reaction

i. Concentration: According to the law of mass action, greater is the concentration of reactant, the more rapidly the reaction proceeds.

ii. Pressure: On increasing the pressure, volume decreases and concentration increases and the rate of reaction increases.

iii. Temperature: It is generally observed that a rise in temperature increases the reaction rate. It has been found that the rate is either doubled or tripled for every 10°C rise in temperature.

iv. Nature of reactants: The rate depends on specific bonds involved and hence on the nature of reactants. Ionic reactions are faster than molecular reactions.

v. Surface area of reactants: An increase in surface area increases the number of molecules undergoing chemical reaction per unit time which increases the rate of reaction. eg. Rate of production of H₂ is more when the granulated zinc is reacted with dil HCl instead of big lumps of zinc.

vi. Catalyst: A catalyst itself doesn’t take part in a reaction but affects the rate of reaction. A positive catalyst increases the rate of reaction whereas the negative catalyst decreases the rate of reaction. Similarly, the rate of biochemical reaction is increased by an enzyme.

Rate law equation

The actual relationship between the concentration of reacting species and the reaction rate is determined experimentally and is given by the expression called rate law equation.
For any hypothetical equation: aA + bB → cC + dD

where m and n are constant numbers or the powers of the concentration of reactants A and B respectively on which the rate of reaction depends.

• Rate of chemical reaction is directly proportional to the concentration of reactants.
• The rate law represents the experimentally observed rate of reaction which depends upon the slowest step of the reaction.
• Rate law cannot be deduced from the relationship for a given equation. It is found from experiment only.

Rate constant

The proportionality constant K in the above rate law equation is called rate constant or velocity constant or specific reaction rate. Let us consider the unit concentration (1 mol L^-1) of each reactant,

Hence the rate constant K is defined as the rate of reaction when the concentration of each reacting species is taken as unity.

• The value of rate constant depends on nature of reactant, temperature and catalyst. It is independent on concentration of reactant.
• Unit of rate constant = [mole/L]^1-n x sec^-1 where n is the order of reaction.

Molecularity of reaction

The number of reacting species (atoms, ions or molecules) present in the rate-determining step. For a one-step reaction, molecularity is the number of reacting species as represented by a balanced chemical equation.
eg. NH₄NO₂ → N₂ + 2H₂O

One molecule is present in this equation. So, its molecularity is 1. i.e. unimolecular reaction.
eg. 2HI → H₂ + I₂

Two molecules are present in this equation. So, its molecularity is 2 i.e. bimolecular reaction. But for the complex reaction that completes in two or more than two steps having a mechanism, the slowest step is the rate-determining step and the number of species present in the rate-determining step is its molecularity. Molecularity may be less than equal to 3.

Order of reaction

The order of a reaction may be defined as the sum of the power of concentration terms raised in an experimentally determined rate law equation.
For a reaction: aA + bB → Product

order with respect to A = m
order with respect to B = n
Total order = m+n
m and n may or may not be equal to a and b.

Difference between molecularity and order of reaction

Types of order of reaction

1. Zero order reaction

The reaction in which the rate of reaction is independent of the concentration of reaction is called a zero-order reaction.
Let us consider a general zero-order reaction. Let a is the initial concentration of a reactant at time t = 0 and after some time t, x amount of product is formed.
A → Product

The rate of the above reaction is given by

\begin{align*} Rate &= K_{o}[A]^{o}\\
or,\ \frac{dx}{dt}&=K_{o}\\
[Differential\ rate\ &eqn\ for\ zero\ order\ rxn.]\\
or,\ \frac{dx}{dt}&=K_{o}(a-x)^{o}\\
or,\ dx&=K_{o}dt\\
Now,\ integrating&\ above\ equation,\\
\int dx&=K_{o}\int dt\\
or,\ x&= K_{o}t+I
\end{align*}

where I is integration constant. To know its value, let us apply the initial condition i.e. when time = 0, the amount of product is also 0 i.e. x=0. Substituting this value in the above equation, we get,

0 = K_o x 0 + I
so, I = 0
Hence,
x = K_ot
K_o = x/t —–(i)

This is the integrated rate law equation for the zero-order reaction.

Characteristics of zero order reactions

a. The unit of zero-order reaction is mol L^-1sec^-1.

b. Half-life period: It is the time during which half of the initial concentration of reactant is converted into product. It is
denoted by t_1/2.
According to the definition, at the half-life period (t_1/2), x = a/2, where a is the initial concentration of reaction.
Substituting this value in equation (i), we get,

\begin{align*} K_{o}&=\frac{a/2}{t_{1/2}}\\
or,\ t_{1/2}&=\frac{a}{2K_{o}}\\
or,\ t_{1/2}\ &\alpha\ a
\end{align*}

Hence, the half-life period of the zero-order reaction is directly proportional to the initial concentration of the reactant.

c. Kinetic plot:
We have,
K_o = x/t
or, x = K_ot
A plot of x versus t gives a straight line passing through the origin. The slope of a straight line is the rate constant of the zero-order reaction.

Examples of zero order reactions

\begin{align*} i.&\ H_{2}+Cl_{2}\xrightarrow{h\nu}2HCl\\
ii.&\ N_{2}O \xrightarrow{Pt} N_{2} + \frac{1}{2}O_{2}\\
iii.&\ 2NH_{2}\xrightarrow{Fe} N_{2} + 3H_{2}\\
iv.&\ 2HI\xrightarrow{Au} H_{2} + I_{2}\\
v.\ &Enzyme\ catalysed\ reaction.
\end{align*}

2. First order reaction

The reaction in which the rate of reaction depends upon one concentration term is called a first-order reaction.
Let us consider a general first-order reaction. Let the initial concentration of the reactant be a mol L^-1 and after time t, x mol of reactant changes into product.

The rate of the above reaction is given by:

\begin{align*} Rate&=K_{1}[A]^{1}\\
\frac{dx}{dt}&=K_{1}(a-x)^{1}\\
\end{align*}

This is the differential rate law equation for a first-order reaction.

\begin{align*} \frac{dx}{(a-x)}&=K_{1}dt\\
Integrating\ &it,\ we\ get,\\
\int \frac{dx}{(a-x)}&=K_{1}\int dt\\
or,\ -ln(a-x)&=K_{1}t+I
\end{align*}

where I is integration constant. To know its value, let us apply the initial condition i.e. when time = 0, the amount of product is also 0 i.e. x=0. Substituting this value in the above equation, we get

ln(a-0) = K₁ x 0 + 1
so, – ln a = I
Hence,

\begin{align*} -ln(a-x) &= K_{1}t - lna\\
or,\ K_{1}t = &lna - ln(a-x)\\
or,\ K_{1}t&=ln\frac{a}{(a-x)}\\
or,\ K_{1}=\frac{2.303}{t}&\log\frac{a}{(a-x)}---(i)
\end{align*}

This is the integrated rate law for the first-order reaction.

Characteristics of first order reaction

a. Unit of first-order reaction:
For the first-order reaction,

\begin{align*} Rate&=K_{1}[A]\\
or,\ mol\ L^{-1}sec^{-1}&=K_{1} [mol L^{-1}]\\
so,\ K_{1} &= sec^{-1}
\end{align*}

Hence, the unit of rate constant for a first-order reaction is sec^-1.

b. Half-life period: It is the time during which half of the initial concentration of reactant is converted into product. It is
denoted by t_1/2.
According to the definition, at the half-life period (t_1/2), x = a/2, where a is the initial concentration of reaction.
Substituting this value in equation i, we get,

\begin{align*} K_{1}&=\frac{2.303}{t_{1/2}}\ log\frac{a}{(a-a/2)}\\
K_{1}&=\frac{2.303}{t_{1/2}}log2\\
t_{1/2}&=\frac{0.693}{K_{1}}
\end{align*}

It shows that the half-life of the first-order reaction is independent of initial concentration.

c. Kinetic plot: The rate constant for the first-order reaction is:

\begin{align*} K_{1}&=\frac{1}{t}ln\frac{a}{(a-x)}\\
or,\ ln&\frac{a}{(a-x)}=K_{1}t
\end{align*}

This equation is in the form of y=mx. If a graph of ln[a/(a-x)] vs. time is plotted, a straight line passing through the origin is obtained.

The above equation can also be expressed as:

\begin{align*} lna - ln(a-x) &= K_{1}t\\
or,\ ln(a-x) &= -K_{1}t + lna
\end{align*}

This is in the form of y = mx + c. If a graph is plotted, a straight line with a negative slope is obtained.

Examples of first order reactions

a. Decomposition of H2O2
2H₂O₂ → 2H₂O + O₂

b. Decomposition of N2O5
2N₂O₅ → 4NO₂ + O₂

c. Decomposition of NH4NO2
NH₄NO₂ → N₂ + 2H₂O

d. Decomposition of SO₂Cl₂
SO₂Cl₂ → SO₂ + Cl₂

e. All radioactive disintegration reactions

Pseudo-order reaction

The reaction which seems to be of higher order kinetics but actually follows the lower order kinetics when one of the reactants is present in excess amount is called pseudo-order reactions.
eg. i. Hydrolysis of ethyl acetate in acidic medium

\underset{ethyl\ acetate}{CH_{3}COOC_{2}H_{5}} + \underset{excess}{H_{2}O} \xrightarrow{HCl}\underset{acetic\ acid}{CH_{3}COOH} + C_{2}H_{5}OH

ii. Hydrolysis of cane sugar

\underset{sucrose}{C_{12}H_{22}O_{11}} + \underset{excess}{H_{2}O} \xrightarrow{invertase}\underset{glucose}{C_{6}H_{12}O_{6}} + \underset{fructose}{C_{6}H_{12}O_{6}}

Collision theory

i. The basic requirement for a reaction to occur is that the reacting species must collide with each other. This is the basis of collision theory for reaction.
ii. The number of collisions that takes place per second per unit volume of the reaction mixture is known as collision frequency (Z).
iii. Every collision does not bring a chemical change. The collisions that actually produce the products are effective collisions. The effective collisions which bring chemical change are few in comparison to the total number of collisions. The collisions that do not form a product are called ineffective collisions i.e. molecules just collide and disperse in different directions with different velocities.
iv. For a collision to be effective the following two barriers are to be cleared.

A. Energy barrier

The minimum amount of energy that the collision molecules must possess to make the chemical reaction occur is known as threshold energy.

There is an energy barrier for each reaction. The reacting species must be provided with sufficient energy to cross the energy barrier.

B. Orientational barrier

The colliding molecules should also have proper orientation so that the odd bonds may break and new bonds are formed. For example,
NO₂ + NO₂ → N₂O₄

During this reaction, the products are formed only when the colliding molecules have proper orientation of the time of the collision. These are called effective collisions.

Main points of collision theory

• For a reaction to occur, there must be collision between the reacting species.
• Only the certain fraction of total number of collision is effective in forming the product.
• For effective collision, the molecules should possess sufficient energy as well as orientation.

Factors affecting rate of reaction using collision theory

i. Temperature: When the temperature is increased, the frequency of collision is increased which causes products to form faster. Lowering the temperature usually slows down the reaction.

ii. Concentration: Having more particles into a fixed volume increases the concentration. This increases collision frequency and leads to a higher rate of reaction.
-Lower concentration: Few collision
-Higher concentration: More collision

iii. Particle size: The total surface area of a solid or liquid of reactant affects the rate of reaction. The smaller the particle size, greater is the surface area for a given mass of particles. This results in an increase in collision frequency and rate of reaction increases.
We can also increase the surface area of the solid by dissolving it. In a solution, particles are separated and more accessible to other reactants.
Another way to increase the surface area of a solid is by grinding it into a fine powder.

Activation energy

The reactants do not convert directly into the products. Whether the reaction is exothermic or endothermic, the energy of the reactant is sufficient to give a product. Hence, energy must be added so that the reactant can cross the energy barrier to give the product. The concept of energy for the reaction system is clearly understood from the figure which is called the energy profile diagram.

The minimum amount of energy that must be supplied to the reactant by the effective collision to cross the energy barrier for giving the product is called activation energy. In the energy profile diagram, energy equivalent to BC represents the activation energy. AB represents the energy associated with reactants.
The minimum amount of energy that must be associated with the reactant molecules to cross the energy barrier for giving the product is called threshold energy. In the above diagram, energy equivalent to AC represents threshold energy.

Threshold energy = Activation energy + Energy of reactant

Activated complex theory

Collision theory is inadequate to explain the rate of a reaction involving complex molecules. In 1935, Henry Eyring and Polyani gave a new theory to explain the dependence of reaction rate on concentration, temperature and other various factors in which activated complex or transition state is formed during transformation of reactant into the product. This theory is called activated complex formation theory or transition state theory or theory of absolute reaction rate.
The main postulates of this theory are as follows:

i. The reactant molecule is transferred into high energy containing intermediate complex before changing into a product and such an intermediate complex associated with high energy is called an activated complex or transition state.

ii. The activated complex is in equilibrium with the reactant.
Reactants ⇌ Activated complex

iii. The activated complex decomposes into products.
Activated complex → Product

iv. The rate of reaction is given by the rate of decomposition of the activated complex.

The above points may be summarized by the following reaction and energy profile diagram.

\underset{Reactant}{A+B}\rightleftharpoons \underset{\substack{Activated\\ complex}}{AB^{*}}\rightarrow Product